There are some really good yoga poses that might help. Do a search for yoga poses and you should come up with some good ones.
I was going to suggest water training as well.
And a stationary bike is probably a lot easier on the knees, too.
Other than that I'm not sure but I wish you much luck! Looks like previous commenter gave you a wonderful link, too.
I like your name. It makes you sound optimistic and we'll be optimistic that there IS something out there for you to do to safely burn calories!
I would think that WATER would be safe, if you have access to a pool (like your local YMCA). Also, cycling is usually good as long as it is not weight-bearing. Most gyms offer indoor classes, even if the weather where you live doesn't permit year-round outdoor riding.
I googled safe exercises for bad knees and came up with this: http://www.prevention.com/cda/article/best-and-worst-exercises-for-bad-knees/ff53323b0b803110VgnVCM20000012281eac____/fitness/fitness.for.your.health/exercises.for.specific.conditions/arthritis/
I have not had time to look at it. Let me know if it helps. And, keep "talking" to us on here! You might also try YOGA. It is gentle on the joints, but a very good form of exercise.
Good luck and hang in there!
P.S. To better clarify my bad shoulders: I had torn rotator cuffs in each of my shoulders...I got one repaired but cant lift but up to 5 pounds and sometimes that hurts. The other one is still torn.